DIFFERENTIAL boundary of the s-region corresponding to the two sides of the barrier (- °°, 0) meet (at s = 0 if the real part of Xx be positive) at an angle 2TrL1, where Lj is the absolute value of the real part of Xj; the same is true for the c-region representing the branch a. The condition that the s-region shall not overlap itself requires, then, Li <1. But, further, we may form an infinite number of branches <r = se2”Xl, cr1 = <re2nXl, • • • in the same way, and the corresponding regions in the plane upon which y2ly1 is represented will have a common point and each have an angle 27rL1; if neither overlaps the preceding, it will happen, if Lj is not zero, that at length one is reached overlapping the first, unless for some positive integer a we have 27raL1 = 27t, in other words L1 = 1 /a. If this be so, the branch = se27r<0iXi will be represented by a region having the angle at the common point common with the region for the branch s; but not altogether coinciding with this last region unless Ax be real, and therefore =+l/a ; then there is only a finite number, a, of branches obtainable in this way by crossing the barrier (- oo, 0). In precisely theX same way, if we had begun by taking the quotient s' = (x-1) 2F( +X2, /4 +A2, 1 + X2, 1-a:)/ F(A, ya, 1 - X2, 1 - x) of the two solutions about x = 1, we should have found that x is not a single-valued function of s' unless X2 is the inverse of an integer, or is zero ; as s' is of the form (As + B)/(Cs + D), A, B, C, D constants, the same is true in our case ; equally, by considering the integrals about x—'x> we find, as a third condition necessary in order that x may be a single-valued function of s, that X—/4 must be the inverse of an integer or be zero. These three differences of the indices, namely, A^ X2, A - /*, are the quantities which enter in the differential equation satisfied by a; as a function of s, which is easily found to be - X^ + ~^—{h-h1-h^)x~x {x-VT' + Wx 22+ P2(x-l) 2, 2where x1=dxlds, etc.; and 7^= 1 - A^, A2=l - X2 ,/i3 = l - (A-ju) . Into the converse question whether the three conditions are sufficient to ensure (1) that the s region corresponding to any branch does not overlap itself, (2) that no two such regions overlap, we have no space to enter. The second question clearly requires the inquiry whether the group (that is, the monodromy group) of the differential equation is properly discontinuous. (See art. Groups.) The foregoing brief account will give an idea of ‘the nature of the function theories of differential equations; it appears essential not to exclude some explanation of a theory intimately related both to such theories and to transformation theories, which is a generalization of Galois’s theory of algebraic equations. We deal only with the application to homogeneous linear differential equations. In general a function of variables x1, x2. • • • is said to be rational when it can be formed from them and the integers 1, 2, 3 • • • by a p t; aa at i'm^e number of additions, subtractions, multiplications, ro ° f divisions. We generalize this definition. Assume a liae °r that an< we have assigned a fundamental series of quantities equation that^ functions of x, formed in whichbyxa itself included, such all quantities finite isnumber of additions, subtractions, multiplications, divisions, and differentiations in regard to x, of the terms of this series, are themselves members of this series. Then the quantities of this series, and only these, are called rational. And by a rational function of quantities p, q, r • • • is meant a function formed from them and any of the fundamental rational quantities by a finite number of the five fundamental operations. Thus it is a function which would be called, simply, rational if the fundamental series were widened by the addition to it of the quantities p, q,r • • • and those derivable from them by the five fundamental operations. A rational ordinary differential equation, with x as independent and y as dependent variable, is then one which equates to zero a rational function of y, the order k of the differential equation being that of the highest differential coefficient yW which enters ; only such equations are here discussed. And such an equation P = 0 is called irreducible when, firstly, being arranged as an inIrreduci- ^ero, ora^ polynomial in y(k this polynomial is not the bility of a PorrnBlctanof other polynomials in yW also of rational rational * > a d, secondly the equation has no solution satisequation. b° a rational equation of lower order. From this it follows that if an irreducible equation P = 0 have one solution satisfying another rational equation Q = 0 of the same or higher order, then all the solutions of P = 0 also satisfy Q = 0. lor from the equation P = 0 we can by differentiation express y(k+i)^ y(k+2). .. ixx terms of x, y, yW, • • •, y(k and so put the function Q rationally in terms of these quantities only. It is sufficient, then, to prove the result when the equation Q = 0 is of the same order as P = 0. Let both the equations be arranged as
EQUATIONS
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integral polynomials in yW ; their algebraic eliminant in regard to y(k) must then vanish identically, for they are known to have one common solution not satisfying an equation of lower order; thus the equationnP = 0 involves Q = 0 for all solutions of P = 0. Now let y( y=a1y(n~1) + • • + any be a given rational homogeneous linear differential equation ; let y^ • ■ yn be n particular functions of x, unconnected by any equation with constant coefficients of the form cly1 + • • + cnyn - 0, all satisfying the differ- va^Jant ential equation; let Vi' ■ Vn he linear functions of functjon y1 • • yn, say 7H=kily1 + - ■ + kinyn, where the constant /or a'° coefficients Ay have a non-vanishing determinant; jjaear write (r;) — A(y), these being the equations of a general equation. linear homogeneous group whose transformations may be denoted by A, B, • • •. We desire to form a2, rational function <p{y), or say <p(±(y)), of y1- ■ yni in which the ri constants Ay shall all be essential, and not reduce effectively to a fewer number, as they would, for instance, if the y^. . yn were connected by a linear equation with constant coefficients. Such a function is in fact given, if the solutions yx- -yn be developable in positive integral powers about x=a, by 0(57) = 9?i + (a: - a)”% + • • • + (x Such a function, V, we call a variant. Then differentiating Y in regard to x, and replacing %(”) by its -1 value d T jdx, and similarly each 2 ax^f”2 ) +N• • + an Nr], we can arrange 2 of d Y/dx • • d Yjdx , where N=» , -1 as a linear func- . tion of the N quantities Vx ■ • yn • • V” ). • r]nin~1), and so^a"f thence by elimination obtain a linear differential equa- eaqnation tion for V of order N with rational coefficients. This we denote by F = 0. Further, eachNof1 ^x • •N_1 •)?« is expressible as a linear function of V, dV jdx • • • d ~2 Y/c?x , with rational coefficients not involving any of the n coefficients A,-,-, since otherwise V would satisfy a linear equation of order less than N, which is impossible, as it involves (linearly) the n2 arbitrary coefficients Ay, which would not enter into the coefficients of the supposed equation. In particular, y^ •N• y1n are Nexpressible rationally as linear functions of «, dujdx, • • • d ~ bi{dJx ~v, where w is the particular function <t>{y'). Any solution W of the equation F = 0 is derivable from functions £x • • fn, which are linear functions of yx • • yn, just as Y was derived from yY ••??»; but it does not follow that these functions •• are obtained from yx • • yn by a transformation of the linear group A, B, • • ; for it may happen that the determinant d(fx • • fn)/^(?/i • • Vu) is zero. In that case £i • • may be called a singular set, and W a singular solution ; it satisfies an equation of lower than the N-th order. But every solution V, W, ordinary or singular, of the equation F = 0, is expressible rationally in terms of w, du/dx, • • • • dN~1cojdxN~1; we shall write, simply, Y = r(w). Consider now the rational irreducible equation of lowest order, not necessarily a linear equation, which is satisfied by w; as 7/x • • y„ are particular functions, it may quite well be of order less than N ; we call it the resolvent equation, suppose it of order p, and denote it by 7(0). Upon it the whole theory turns. In the first place, as 7(1;) = 0 is satisfied by the solution w of F = 0, all the solutions of y(v) are solutions F = 0, and are therefore rationally expressible by w ; any one may then be denoted by r(w). If this solution of F = 0 be not singular, it corresponds to a transformation A of the linear group (A, B, • • )> effected upon yx • ■ yn- The coefficients Ay of this transformation follow from the expressions before mentioned for ??i • • ??« in terms of Y, dV/dx, d2Yjdx?, • • by substituting Y — r{bi); thus they depend on the p arbitrary parameters which enter into the general expression for the integral of the equation 7(1;) = 0. Without going into further details, it is then clear enough that the resolvent equation, being irreducible and such that any solution is expressible rationally, with p parameters, in terms of the solution «, enables us to define a linear homogeneous group of transformations of yx • • yn depending on p parameters ; and every operation of this (continuous) group corresponds to a rational transformation of the solution of the resolvent equation. This is the group called the rationality group, or the group of transformations of the original homogeneous linear differential equation. The group must not be confounded with a subgroup of itself, the monodromy group of the equation, often called simply the group of the equation, which is a set of transformations, not depending on arbitrary variable parameters, arising for one particular fundamental set of solutions of the linear equation (see art. Groups). The importance of the rationality group consists in three propositions. (1) Any rational function of yx • • yn which is unaltered in value by the transformations of the group can be written in rational form. (2) Ifany rational function be changed Tne^jin^ in form, becoming a rational function of yx ■ • yn, a theorem transformation of the group applied to its new form will ar(j leave its value unaltered. (3) Any homogeneous linear tbe transformation leaving unaltered the value of every ratjon, rational function of yx- ■ yn which has a rational value, ajjty belongs to the group. It follows from these that any groupt group of linear homogeneous transformations having the properties (I) (2) is identical with the group in question. It is clear that with these properties the group must be of the greatest S. III.— 58
